Introduction

1. Definition of a Crystal and Crystallization
2. Crystal Elements:
   • Crystal Faces
   • Edges
   • Solid Angles
   • Interfacial Angles
3. Elements of Symmetry:
   • Axes of Symmetry
   • Plane of Symmetry
   • Centre of Symmetry
4. Miller Indices
5. Law of Symmetry
6. Crystallographic Systems:
   • Cubic System
   • Tetragonal System
   • Hexagonal System
   • Trigonal System
   • Orthorhombic System
   • Monoclinic System
   • Triclinic System

Definition of a Crystal and Crystallization

• Crystal: A solid material whose atoms are arranged in a highly ordered, repeating pattern.
• Crystallization: The process by which a solid forms, with atoms or molecules aligning in a structured, repeating pattern.

Crystal Elements

• Crystal Faces: The flat surfaces that bound a crystal.
• Edges: The line where two faces of a crystal meet.
• Solid Angles: The point where three or more faces of a crystal meet.
• Interfacial Angles: The angle formed between two crystal faces.

Crystallographic Axes

Crystallographic Axes and Their Names

• Crystals are described using three or four imaginary lines known as crystallographic axes.
• These axes are labeled as a, b, c:
  • a-axis: Horizontal axis (length)
  • b-axis: Horizontal axis (width, perpendicular to a)
  • c-axis: Vertical axis (height)
• In some systems, a fourth axis is introduced (used in hexagonal and trigonal systems).

Angles Between the Crystallographic Axes

• The angles between the axes define the crystallographic system:
  • α: Angle between b and c axes
  • β: Angle between a and c axes
  • γ: Angle between a and b axes

Importance of Crystallographic Axes

• These axes are used to define the orientation of crystal faces and the overall symmetry of the crystal.
• The angles between the axes help determine which crystallographic system a mineral belongs to.

Crystallographic Systems and Their Axes

1. Cubic: a = b = c; α = β = γ = 90°
2. Tetragonal: a = b ≠ c; α = β = γ = 90°
3. Orthorhombic: a ≠ b ≠ c; α = β = γ = 90°
4. Hexagonal: a = b ≠ c; α = β = 90°, γ = 120°
5. Trigonal: a = b = c; α = β = γ ≠ 90°
6. Monoclinic: a ≠ b ≠ c; α = γ = 90°, β ≠ 90°
7. Triclinic: a ≠ b ≠ c; α ≠ β ≠ γ ≠ 90°

Elements of Symmetry

• Axes of Symmetry: Imaginary lines through a crystal about which it can be rotated and appear the same.
• Plane of Symmetry: A plane that divides a crystal into two equal, mirror-image halves.
• Centre of Symmetry: A point within the crystal where any line drawn through it intersects equivalent points on opposite sides.

Law of Symmetry

• Law of Symmetry: States that every crystal exhibits a certain level of symmetry, including rotational, mirror, and inversion symmetries.

Hermann-Mauguin (International) Symbols

Miller Indices

• Miller indices are used to denote the orientation of crystal faces and are represented by three numbers (h, k, l) that define the intercepts of a face with the crystallographic axes.